Hi Captain,
I came out to one of my parents not too long ago (I will tell me other parent next time I see them in person), and I completely blindsided her. Still she was really accepting! Speaking as a child and not a parent, they generally have this whole "unconditional love" thing? Some do anyway, I am one of the lucky ones.
No one can tell you the best way to come out to anyone as there really is no such thing, everyone is different and we all come from and live in different circumstances. I'm seeing a psychologist and am planning on working on how to do it better than last time when I come out to the rest of the family and my friends.
One thing I can recommend though is to have some information handy to give to them regarding what it is to be transgendered, it will help A LOT. Whoever you come out to will have questions, its generally something that people don't know a lot about, my Mum works in an industry where she has some contact with trans people and she still has questions and major concerns for me.
If you can give them a brouchure, websites, I went for a USB with a stack of pdf's from the NSW GenderCentre (an Australia website,
www.gendercentre.org.au) which has a lot of "resource kits" which have all sorts of information in them, definitions and terminology of what is a transgender, transsexual, the differences between them etc. with one specifically for family and friends to educate them (it does have some rather specific information relevant to the state it is in in Australia so if you're in the USA for instance it will be different ie. contact numbers, support groups). The great thing about a USB is that they can pop it in a computer, any computer, and look at it without thinking "what if someone finds it in web browsing history?" or do it on a laptop or computer without internet connection.
Otherwise I would talk with a experienced psychologist, psychiatrist or counselor who can talk through things with you.
Ultimately we have no idea what another person is feeling or will feel but we can try anticipate them, but you may be completely wrong!
Who knows you might do it before your ticker runs up!
Good luck though! Let us know how you go